# Primitive-Fock Interaction Layer

- ID: 2188
- Canonical URL: https://synthcore.org/idea/2188/primitive-fock-interaction-layer
- API JSON: https://synthcore.org/api/idea/2188.json
- API Markdown: https://synthcore.org/api/idea/2188.md
- Verification status: unverified
- Source: [arXiv:2608.12492](https://arxiv.org/abs/2608.12492)
- Category: architecture
- Solves: accuracy, generalization, scalability
- ML areas: mlp, vae, regularization, embedding
- Math tags: probability, functional-analysis, tensor-decomposition, linear-algebra
- Ratings: usefulness 6/10; difficulty 6/10; novelty 7/10

## Idea description

Replace an unconstrained q-way polynomial or tensorized feature layer with separate decomposable and primitive interaction channels. The decomposable channel models interactions explainable as products of lower physical-weight feature blocks, while the primitive channel captures residual factors that cannot be represented by those products. This should reduce redundant high-order parameters and provide a controllable inductive bias for compositional or disentangled representations.

## Mathematical statement

The paper fixes a graded Hilbert space E=E_1\oplus\cdots\oplus E_q, where E_s is the feature space with physical weight s. For an occupation vector \bm{m}=(m_1,\ldots,m_q), its physical weight is \operatorname{wt}(\bm{m})=\sum_{s=1}^q s m_s and its ordinary terminal Gaussian order is |\bm{m}|=\sum_{s=1}^q m_s. The weighted-Fock sector of total physical weight q is \mathcal{W}_q(E)=\bigoplus_{\operatorname{wt}(\bm{m})=q} I_{|\bm{m}|}(\widehat{\bigotimes}_{s=1}^q E_s^{\odot m_s}), where E_s^{\odot m_s} is the symmetric tensor power and I_k denotes the k-th Wiener integral in the terminal Gaussian space. For an input feature vector, the decomposable subspace is generated by products of lower-weight feature blocks; the primitive subspace is its orthogonal complement. Implement the empirical projection P_{\mathrm{dec}} onto the span of product features and P_{\mathrm{prim}}=I-P_{\mathrm{dec}}. The primitive component is the mathematically canonical residual factor, up to graded orthogonal transformations. In the neural layer, use this residual split for learned polynomial interactions and penalize correlation between the two channels.

## Key formulas

- $$\lvert\bm{m}\rvert=\sum_{s=1}^{q}m_s,\qquad \operatorname{wt}(\bm{m})=\sum_{s=1}^{q}s m_s,$$
- $$\mathcal{W}_{q}(E)=\bigoplus_{\operatorname{wt}(\bm{m})=q}I_{\lvert\bm{m}\rvert}\left(\widehat{\bigotimes}_{s=1}^{q}E_s^{\odot m_s}\right),$$
- $$P_{\mathrm{prim}}=I-P_{\mathrm{dec}},\qquad h_{\mathrm{dec}}=P_{\mathrm{dec}}h,\qquad h_{\mathrm{prim}}=P_{\mathrm{prim}}h,$$
- $$\mathcal{L}=\mathcal{L}_{\mathrm{task}}+\lambda_{\perp}\frac{1}{B}\sum_{i=1}^{B}\left\langle h_{\mathrm{dec}}^{(i)},h_{\mathrm{prim}}^{(i)}\right\rangle^2+\lambda_{\mathrm{prim}}\frac{1}{B}\sum_{i=1}^{B}\left\|h_{\mathrm{prim}}^{(i)}\right\|^2\mathbf{1}\{\text{primitive channel is unnecessary}\}.$$

## Implementation notes

Integrate this as a replacement for a degree-q polynomial feature layer inside a small MLP, VAE encoder, or representation-learning model. First choose q=2 or q=3 and split the incoming embedding x into physical-weight blocks u_1,...,u_q with dimensions d_1,...,d_q. Construct all weight-q decomposable products: for q=2 use products u_1\otimes u_1 and u_2; for q=3 use u_1\otimes u_1\otimes u_1, u_1\otimes u_2, and u_3. Flatten or contract each product with learned matrices to obtain a feature matrix D in R^{B x r}. Construct an independent learned residual interaction H in R^{B x r_p}, for example with a bilinear or trilinear map from x. Estimate the decomposable projector from the current minibatch using ridge least squares: A=(D^T D+epsilon I)^{-1}D^T H, then H_dec=D A and H_prim=H-H_dec. Stop gradients through A initially, or update an exponential-moving covariance estimate for greater stability. Concatenate H_dec and H_prim before the output head. The paper's mathematics determines the weighted-sector indexing and orthogonal primitive/decomposable split; D, H, and the empirical projector are estimated from minibatch features. For the first experiment, use a two-layer width-256 MLP on synthetic regression with a target containing a known product term x_1x_2 and an independent latent interaction, then test on a tabular dataset with engineered feature crosses. Compare against a standard MLP and an unconstrained degree-2 polynomial layer at equal parameter count. Measure test error, performance versus width, primitive/decomposable correlation, and accuracy when product features are frozen. Success means lower error or equal error with fewer high-order parameters, plus near-orthogonality and reduced feature redundancy.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.